Limits of Solutions to the Relativistic Euler Equations for Modified Chaplygin Gas by Flux Approximation
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Limits of Solutions to the Relativistic Euler Equations for Modified Chaplygin Gas by Flux Approximation Yu Zhang1,2
· Hanchun Yang1
Received: 28 January 2019 / Accepted: 9 August 2019 © Springer Nature B.V. 2019
Abstract We study the Riemann problem and flux-approximation limits of solutions to the relativistic Euler equations with the state equation for modified Chaplygin gas. The limits of Riemann solutions for the relativistic modified Chaplygin gas equations and the corresponding flux-approximation system are discussed when the pressure term and flux-perturbation parameters tend to zero. It is rigorously proved that, as the pressure and flux approximations vanish respectively, any two-shock-wave Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations, and the intermediate density between them tends to a weighted δ-measure that forms a delta shock wave. Correspondingly, any two-rarefaction-wave solution becomes two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution. Keywords Relativistic Euler equations · Riemann problem · Delta shock wave · Vacuum · Vanishing pressure limit · Flux approximation · Modified Chaplygin gas · Lorentz transformation
This work is supported by the NSF of China (11361073), Yunnan Applied Basic Research Projects (2018FD015) and the Scientific Research Foundation Project of Yunnan Education Department (2018JS150).
B Y. Zhang
[email protected] H. Yang [email protected]
1
Department of Mathematics, Yunnan University, Kunming 650091, P.R. China
2
Department of Mathematics, Yunnan Normal University, Kunming 650500, P.R. China
Y. Zhang, H. Yang
1 Introduction We focus on the relativistic Euler equations modeling the conservation of energy and momentum in special relativity [12, 13] ⎧ v v2 ⎪ 2 2 ⎪ + ρ + p + ρc = 0, p + ρc ⎪ ⎨ c2 (c2 − v 2 ) c2 − v 2 x t (1.1) ⎪ v v2 ⎪ 2 2 ⎪ ⎩ p + ρc 2 + p + ρc 2 + p = 0, c − v2 t c − v2 x where ρ, v and p represent the proper energy density, particle speed and pressure, respectively, c the speed of light. For the isentropic flow, the pressure is a scaled function on density. The system (1.1) describes the dynamics of plane waves in special relativity fluids in a two-dimensional Minkowski space-time (x 0 , x 1 ) div T = 0, where
T ij = p + ρc2 ui uj + pηij
denotes the stress-energy tensor for the fluid, and all indices run from 0 to 1 with x 0 = ct , ηij = ηij = diag(−1, 1) is the flat Minkowski metric, u the 2-velocity of the fluid particle, and ρ the mass-energy density of the fluid as measured in units of mass in a frame moving with the fluid particle. Until now, the system (1.1) has been extensively investigated by mathematics and physics researchers because of the high importance and extreme complexity of the system itself. For example, as regards the polytropic gas p = κ 2 ρ γ (γ > 1), Chen [3] analyzed the properties of elementary waves, and solved the Riemann problem and Cauchy problem. Specifically, when γ = 1, both t
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